Transformations, Squares and Vertexes
by ntepAC
Summary: Leonardo's at it again... Yet another PreCalc assignment. Gotta love Leo, right?


_A/N: Leonardo's at it again…_

_ Ezio Auditore da Firenze and Assassin's Creed (I, II and Brotherhood) belongs to Ubisoft and Ubisoft Montreal, and Leonardo da Vinci belongs to himself, still. I do not own anything in this story, nor do I even really own this story… I can't claim owning the story without claiming owning the characters, now can I?_

_ Alright, so this one kinda confused me a little more than helped, but I still like it! ^_^ Hope it doesn't confuse you, my readers! (ignore any and all grammatical/mathematical errors, I'll fix them when I get around to it, lol) Thanks for reading!_

**Transformations, Vertexes, and Squares**

"Ezio, I'm bored."

Ezio Auditore's eyes flashed over to the moaning artist, then back at his book.

The artist blinked, then moaned again, "Did you hear me? I'm BORED."

Ezio closed his book. "You always harp at me to read, and when I finally am, you won't leave me alone…" He sighed, clenching the bridge of his nose. "Get a book, Leonardo. Read."

"I wanna teach you something!" Leonardo da Vinci stood up from the chair and smiled innocently.

Ezio studied the artist's face. Leonardo's eyes twinkled with delight. "Like what?"

"Remember when I taught you about transformations?"

"Vaguely, but yes." Ezio growled.

The twinkle disappeared. "Well, I want to teach you something else…"

Ezio tried to keep his face as blank as possible, trying to not show he felt sorry for growling. "Alright."

Leonardo probably would've jumped up in the air and clicked his heels in joy if it wasn't for his injured leg.

"How'd you injure your leg, Leonardo?" Ezio just noticed the injury.

"Oh," Leonardo looked at his leg. "Eh, an experiment. Anyhow," He limped over to the table nearby in his workshop. "Ezio, come here."

Ezio sat down his book and followed the artist over to the table.

Leonardo pulled out some graph paper, and some pencils.

"Where did you get graph paper?"

"Part of the experiment…" Leonardo said quickly before hushing Ezio.

Ezio roiled his brows and shook his head. "Sometimes, you confuse me."

"I aim to!" Leonardo smiled. "So, let's see… let's review transformations."

Ezio kept a groan at bay. "_Va bene_."

Leonardo wrote down on the top of the paper a function.

_f(x) = -1/3(2x+3)__2__+4_

"So, tell me the transformations."

Ezio stared at the function. "Er… a is negative one-third, b is two, c is negative three, and d is positive four. So the graph will… shrink vertically by a factor of one third. It will shrink horizontally by a factor of one half…"

"What about that negative with a?" Leonardo reminded.

"Oh yeah, so it will flip over the x axis. The graph will shift to the left three units and up four units."

"Good! See, you do remember this!" Leonardo smiled encouragingly.

Ezio nodded. "I s'pose I do."

"Alright! So, let's see… Completing the square."

A square popped into the assassin's mind, missing a side. He roiled his brows.

"Just wait for it, Ezio." Leonardo nodded. "Those functions we were using to transform those graphs, the ones with x-squared, those were quadratic. So with completing the square, we will be focusing on the x-squared parent graph."

Ezio nodded. "Alright."

"Alright, so, you remember a, b, x minus c, plus d, right?"

"Yeah."

"Okay, so let's get rid of b."

Ezio's mind subconsciously noted the missing side on his mental square. "We gotta find b?"

"No, we don't need it now."

Ezio roiled his brows again.

"Ezio, what are you thinking?"

"A square with a missing side."

Leoanrdo roiled his brows this time, confused. "A square with… huh whah? No, forget that!"

The three sided square disappeared.

"Okay. So, remind yourself again what 'a', 'c', and 'd' do."

Ezio sighed. "I have a book I could be reading, y'know?"

"Yeah, but this is more fun!" Leonardo smiled.

Ezio sighed again. "That's debatable." He smiled and continued before Leonardo could interject. "So if 'a' is bigger than one, then a vertical stretch happens, by what ever factor 'a' is."

Leonardo began rummaging through his stuff on the table.

"Don't you dare get a rubber band… If it is smaller than one, then it's a vertical shrink." Ezio smirked as Leonardo tried to find an excuse to cover his rummaging.

"Good." Leonardo pulled out more pencils.

"'c' says how many units and what direction the graph moves. Positive it moves to the left, negative to the right."

"Remember, that's when it's in the parenthesis that holds true, because in the template, c is negative."

"Okay." Ezio nodded. "And 'd' does the same thing, but this time up or down. Positive is up, negative is down."

"Good. You have good retention."

"Well, I need it for what I do, you know that."

"Overlooked." Leonardo limped around the table to the graph paper. "So here's how you complete the square. Let's take y = x2 - 4x + 5." He wrote the equation on the paper.

_y = x__2 __- 4x + 5_

"We group the first two terms together." Leonardo wrote parenthesis around the x2 and the negative 4x.

_y = (x__2 __- 4x) + 5_

"What's a term?"

"Anything separated by an addition or subtraction sign. Like x-squared is a term, four x is also a term, and five is a term."

"Oh. So two x squared would also be a term?"

"_Ma certo_!" Leonardo exclaimed.

"_Va bene._" Ezio nodded.

"Now, we need to create a perfect trinomial square, which is three terms added together simplified. Like the quantity x minus three squared would be x minus three squared times itself. And when you multiply each quantity you'll get, when simplified that is, x squared minus six x plus nine. Make any sense?" Leonardo looked up at Ezio.

"Uh… Sort of… Continue, I'm sure I'll figure it out."

"Alright." Leonardo nodded. "Anyhow, so after grouping the first two terms together, we leave a space after the four x in the parenthesis to add our new perfect trinomial square."

_y = (x__2 __- 4x + _ ) + 5_

"And how we find that is we take half of the coefficient of the x term, which would be two, and we square it, so we will get positive four."

_y = (x__2 __- 4x + 4 ) + 5_

"BUT! To make sure we keep the equality of the equation, we must also subtract four outside the parenthesis."

This Ezio seemed to understand, because his face lit up for a second.

Leonardo noted the subtle change and wrote in the negative four.

_y = (x__2 __- 4x + 4 ) + 5 – 4_

"Simplify it Eizo."

Ezio looked at the equation. "Uh… we change the five minus four to one."

"Good!"

_y = (x__2 __- 4x + 4 ) + 1_

"To simplify the quantity you factor it. Remember the negative two?"

"No, you kinda overlooked it."

"No, I told you we take half of the coefficient of the x."

"Oh."

Leonardo looked at the paper. "Okay, perhaps I did… But anyhow, half of negative four is negative two."

"Okay. What do we do with it?" Ezio asked.

"We write this."

_(x - 2)__2__ + 1_

"All we have to do to find 'c' of the square is take that half we found earlier and write it before we squared it." Leonardo explained, pointing to several elements on the paper.

"Ah, okay." Ezio nodded. "And so that x minus two squared will get us back to what we had before?"

"Yes!" Leonardo nodded.

Ezio stared at Leonardo. "There's more, isn't there?"

Leonardo nodded. "Always."

The assassin sighed. "Okay, what now?"

"What do we do when we the leading coefficient isn't one?"

"Something complicated?" Ezio joked.

Leonardo overlooked the comment and wrote something new on the back of the paper. "All we add to the process is factoring out the leading coefficient. Here, complete the square of this one."

_y = -2x__2__ + 6x – 7_

"Alright." Ezio took the paper and pencil. "We write parenthesis here, and here."

_y = ( -2x__2__ + 6x) – 7_

"What did you say, factor out the two?" Eizo looked up.

"_Si. _You know how to do that?"

"I think so…" Ezio wrote down on the paper again.

_y = -2 (x__2__ - 3x) – 7_

Leonardo nodded as he watched Ezio work the problem.

"We take half of the coefficient of x, and square it, so that would be negative three halves squared… nine fourths… Gotta keep the problem equal…"

_y = -2 (x__2__ - 3x + 9/4) - 7 - (-2)(9/4)_

"Gah… fractions…" Ezio growled under his breath.

"Hey, fractions aren't that bad." Leonardo smirked.

"Again, that's debatable." Ezio looked up at the artist.

"Either way, you're doing good. Keep going."

Ezio wrote more on the paper. "So… we need the negative three halves here, and we need to simplify everything… negative two times nine fourths is negative nine halves… which is negative five halves when you add fourteen halves to it… Aha! I think this is it!" Ezio pointed to something on the paper.

_y = -2(x - 3/2)__2__ - 5/2_

Leonardo nodded. "Good! That's it! You've got it Ezio!"

Ezio nodded, smiling.

"See? Doesn't it feel good?"

"Sort of…" He nodded, agreeing.

"Ready for the next thing?"

The smile faded. "There's _more_?"

"Just one more thing, I promise."

"Alright, what is it?" Ezio looked at the paper. "I think you need to get more of this paper, where ever you got it from."

"Eh, later." Leonardo pulled out another piece of graph paper. "So the vertex form is next. Let's take the vertex form of the first problem we did."

_y = (x - 2)__2__ + 1_

"The what form?"

"Oh, didn't I mention this was the vertex form?" Leonardo looked shocked.

"No, you didn't."

"Oh…" He said, deflated. "Well, this is the vertex form, what we found through the completing of the square method." The artist seemed peppy, as normal, again.

"Okay."

"What template did we work with, remind yourself."

"Uh… 'a', quantity 'x' minus 'c' squared, plus 'd'."

"Good, and one more thing to add to your arsenal is that 'c' is also 'h', and 'd' is 'k', so?"

"'a', quantity 'x' minus 'h' squared, plus 'k'?"

"_Si_! _Bene_! And the vertex is only h and k. 'a' is positive in our vertex form…"

"There isn't an 'a'."

"Exactly. It's zero, so it's technically positive."

"But, couldn't it also be negative?"

"Shht!" Leonardo growled at Ezio. "Positive." He said sternly.

"Okay, okay…" Ezio nodded. "Zero's positive… so what?"

"Since 'a' is positive, the parabola opens up, and the low point of the parabola is two, one."

"Oh! Okay…"

"What did what I said just mean?"

"It opens up, and the vertex of the parabola is at two, one on the graph. Two would be on the x axis, and one would be on the y axis."

"Good. You do understand this! See?" Leonardo smiled at Ezio, but stopped when he saw the assassin stand up and walk over to his book, picking it up.

"Thank you Leonardo, but now I would like to read." Ezio sat down and began searching for where he was.

Leonardo huffed. "You'll need this someday!"

"Leonardo, I am an assassin. I kill people. Why would I need to complete the square?" Ezio leaned forward and looked at Leonardo.

"Oh… right…" Leonardo deflated again. "Um… How's about I just stick to the codex pages, and you stick to defeating the Templars?"

"Sounds good to me." Ezio nodded, and looked back at his book again. "Wait, how did you get the graph paper?" Ezio looked up.

Leonardo smiled. "Time travel."

Ezio looked incredulous. "Okay… yeah, keep to the codex."

Leonardo nodded. "Will do."


End file.
